Average Error: 10.7 → 0.5
Time: 4.5s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{z - a}\]
↓
\[\left(y \cdot \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{z - a} \cdot \sqrt[3]{z - a}}\right) \cdot \frac{\sqrt[3]{z - t}}{\sqrt[3]{z - a}} + x\]
x + \frac{y \cdot \left(z - t\right)}{z - a}
↓
\left(y \cdot \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{z - a} \cdot \sqrt[3]{z - a}}\right) \cdot \frac{\sqrt[3]{z - t}}{\sqrt[3]{z - a}} + x
double f(double x, double y, double z, double t, double a) {
        double r646693 = x;
        double r646694 = y;
        double r646695 = z;
        double r646696 = t;
        double r646697 = r646695 - r646696;
        double r646698 = r646694 * r646697;
        double r646699 = a;
        double r646700 = r646695 - r646699;
        double r646701 = r646698 / r646700;
        double r646702 = r646693 + r646701;
        return r646702;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r646703 = y;
        double r646704 = z;
        double r646705 = t;
        double r646706 = r646704 - r646705;
        double r646707 = cbrt(r646706);
        double r646708 = r646707 * r646707;
        double r646709 = a;
        double r646710 = r646704 - r646709;
        double r646711 = cbrt(r646710);
        double r646712 = r646711 * r646711;
        double r646713 = r646708 / r646712;
        double r646714 = r646703 * r646713;
        double r646715 = r646707 / r646711;
        double r646716 = r646714 * r646715;
        double r646717 = x;
        double r646718 = r646716 + r646717;
        return r646718;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original10.7
Target1.4
Herbie0.5
\[x + \frac{y}{\frac{z - a}{z - t}}\]

Derivation

  1. Initial program 10.7

    \[x + \frac{y \cdot \left(z - t\right)}{z - a}\]

  2. Simplified3.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{z - a}, z - t, x\right)}\]
  3. Using strategy rm

  4. Applied fma-udef3.1

    \[\leadsto \color{blue}{\frac{y}{z - a} \cdot \left(z - t\right) + x}\]
  5. Using strategy rm

  6. Applied div-inv3.1

    \[\leadsto \color{blue}{\left(y \cdot \frac{1}{z - a}\right)} \cdot \left(z - t\right) + x\]

  7. Applied associate-*l*1.6

    \[\leadsto \color{blue}{y \cdot \left(\frac{1}{z - a} \cdot \left(z - t\right)\right)} + x\]

  8. Simplified1.5

    \[\leadsto y \cdot \color{blue}{\frac{z - t}{z - a}} + x\]
  9. Using strategy rm

  10. Applied add-cube-cbrt2.0

    \[\leadsto y \cdot \frac{z - t}{\color{blue}{\left(\sqrt[3]{z - a} \cdot \sqrt[3]{z - a}\right) \cdot \sqrt[3]{z - a}}} + x\]

  11. Applied add-cube-cbrt1.9

    \[\leadsto y \cdot \frac{\color{blue}{\left(\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}\right) \cdot \sqrt[3]{z - t}}}{\left(\sqrt[3]{z - a} \cdot \sqrt[3]{z - a}\right) \cdot \sqrt[3]{z - a}} + x\]

  12. Applied times-frac1.9

    \[\leadsto y \cdot \color{blue}{\left(\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{z - a} \cdot \sqrt[3]{z - a}} \cdot \frac{\sqrt[3]{z - t}}{\sqrt[3]{z - a}}\right)} + x\]

  13. Applied associate-*r*0.5

    \[\leadsto \color{blue}{\left(y \cdot \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{z - a} \cdot \sqrt[3]{z - a}}\right) \cdot \frac{\sqrt[3]{z - t}}{\sqrt[3]{z - a}}} + x\]

  14. Final simplification0.5

    \[\leadsto \left(y \cdot \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{z - a} \cdot \sqrt[3]{z - a}}\right) \cdot \frac{\sqrt[3]{z - t}}{\sqrt[3]{z - a}} + x\]

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, A"
  :precision binary64

  :herbie-target
  (+ x (/ y (/ (- z a) (- z t))))

  (+ x (/ (* y (- z t)) (- z a))))